Costat-Time Distributed Domiatig Set Approximatio* Fabia Kuh Departmet of Computer Sciece ETH Zurich 8092 Zurich, Switzerlad kuh @ if.ethz.ch Roger Wattehofer Departmet of Computer Sciece ETH Zurich 8092 Zurich, Switzerlad wattehofer @ if.ethz.ch ABSTRACT Fidig a small domiatig set is oe of the most fudametal problems of traditioal graph theory. I this paper, we preset a ew fully distributed approximatio algorithm based o LP relaxatio techiques. For a arbitrary parameter k ad maximum degree A, our algorithm computes a domiatig set of expected size O(kA 2/k log AIDSoPTI) i O(k 2) rouds where each ode has to sed O(k2A) messages of size O(logA). This is the first algorithm which achieves a o-trivial approximatio ratio i a costat umber of rouds. Categories ad Subject ]Descriptors F.2.2 [Aalysis of Algorithms ad Problem Complexity]: Noumerical Algorithms ad Problems--computatios o discrete structures; G.2.2 [Discrete Mathematics[: Graph Theory--graph algorithms; G.2.2 [Discrete Mathematics]: Graph Theory--etwork problems Geeral Terms Algorithms, Theory Keywords Domiatig Sets, Approximatio Algorithms, Distributed Algorithms, Liear Programmig, Ad-Hoc Networks. INTRODUCTION I a graph, a domiatig set is a subset of odes such that for every ode v either a) v is i the domiatig set or b) a direct eighbor of v is i the domiatig set. The miimum domiatig set (MDS) problem asks for a domiatig set of *The work preseted i this paper was supported (i part) by the Natioal Competece Ceter i Research o Mobile Iformatio ad Commuicatio Systems (NCCR-MICS), a ceter supported by the Swiss Natioal Sciece Foudatio uder grat umber 5005-67322. Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without f provided that copies arc ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, or rcpublish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. PODC'03, July 3-6, 2003, Bosto, Massachusetts, USA. Copyright 2003 ACM -583-708-7K~3/0007...$5.00. miimum size. MDS ad the closely related miimum set cover problem are two of the first problems that have bee show to be NP-hard [8, 2]. I this paper, we preset a distributed approximatio algorithm for MDS. I computer etworks it is ofte desirable to have a domiatig set i order to eable a hierarchical structure i which the members of the domiatig set provide a service for their eighbors. A particular applicatio ca be foud i the fast growig field of mobile ad-hoc etworks. I mobile ad-hoc etworks, wireless devices (called odes) commuicate without statioary server ifrastructure. Whe sedig a message from oe ode to aother, itermediate etwork odes have to serve as reuters. Although a umber of iterestig suggestios have bee made, fidig efficiet algorithms for the routig process remais the most importat problem for adhoc etworks. Sice the topology of a ad-hoc etwork is costatly chagig, routig protocols for ad-hoc etworks differ sigificatly from the stadard routig schemes which are used i wired etworks. Oe effective way to improve the performace of routig algorithms is by groupig odes ito clusters. The routig is the doe betwee clusters. The most basic method for clusterig is by calculatig a domiatig set. Oly the odes of the domiatig set (the 'cluster heads') act as reuters, all other odes commuicate via a eighbor i the domiatig set. Betwee traditioal wired etworks ad mobile ad-hoc etworks two mai distictios ca be made: ) typically wireless devices have much lower badwidth tha their wired couterparts ad 2) wireless devices are mobile ad therefore the topology of the etwork chages rather frequetly. As a cosequece, distributed algorithms which ru o such devices should have as little commuicatio as possible ad they should ru as fast as possible. Both goals ca oly be achieved by developig algorithms requirig a small umber of commuicatio rouds oly (ofte called local algorithms). So far, the oly algorithm which achieves a otrivial approximatio ratio--o(a)--i a otrivial umber of rouds--o(diam(g))--for MDS was developed by Jia, Rajarama, ad Suel [0]. I expectatio, their algorithm achieves a O(logA)-approximatio while the umber of rouds is O(loglogA) with high probability. I this paper, we preset the first distributed MDS algorithm which achieves a otrivial approximatio ratio i a costat umber of rouds. Precisely, for a arbitrary parameter k, i O(k 2) rouds, we achieve a expected approximatio ratio of O(kA 2/k log A). All messages are of size O(log A). 25
The paper is structured i the followig way. Sectio 2 gives a overview over relevat previous work, Sectio 3 itroduces some otatio as well as some well-kow facts, ad i Sectios 4 ad 5 the domiatig set algorithm is developed. Thereby Sectio 4 itroduces the fractioal domiatig set problem (LP relaxatio) ad presets a algorithm to deduce a domiatig set from a solutio to the fractioal variat of the problem, whereas Sectio 5 shows how to approximate the fractioal domiatig set problem by meas of a distributed algorithm. The paper is cocluded i Sectio 6. 2. RELATED WORK The problem of fidig small domiatig sets i a graph ad the closely related problem of fidig small set covers has extesively bee studied over the last 30 years. The problem of fidig a miimum domiatig set has bee prove to be NP-hard i [8, 2]. The best kow approximatio is achieved by the greedy algorithm [, 4, 8]. As log as there are ucovered odes, the greedy algorithm picks a ode which covers the biggest umber of ucovered odes ad puts it ito the domiatig set. It achieves a approximatio ratio of I A where A is the highest degree i the graph. Uless the problems of NP ca be solved by determiistic 0 gl s ) algorithms, this is the best possible up to lower order terms [6]. For the related problem of fidig small coected domiatig sets, a similar approach is show to be a (l A + O())-approximatio i [9]. For the distributed costructio of domiatig sets, several algorithms have bee developed. I [3] a algorithm which calculates a domiatig set of size at most /2 i O(log*) rouds has bee proposed. [9] presets a (coected) domiatig set algorithm which rus i a costat umber of rouds. Noe of those algorithms achieves a o-trivial asymptotic boud o the approximatio ratio. Note that O(A) is trivial sice the set V of all odes of G forms a domiatig set of size at most (A + ) times the size of a optimal oe. The first algorithm which achieves a otrivial approximatio ratio i less tha O(diam(G)) rouds was preseted i [0]. The expected approximatio ratio is asymptotically optimal--o(log A)--ad the algorithm termiates after O(loglog A) rouds with high probability. The algorithm of [0] is related to the parallel set cover algorithms i [3, 6], which achieve O(log A) approximatios i polylogarithmic time. For the coected domiatig set problem, a distributed algorithm which also achieves a approximatio ratio of O(log A) i a polylogarithmic umber of rouds has bee preseted i [5], recetly. I our algorithm, we first solve the LP relaxatio--a positive liear program---of MDS. Parallel ad distributed algorithms for positive liear programmig have bee studied i [5] ad [2], respectively. I polylogarithmic time they both achieve a ( + e)-approximatio for the liear program. For ad-hoc etworks, the (coected) domiatig set problem has also bee studied for special graphs. I particular for the uit disk graph a umber of publicatios have bee writte (e.g. [, 7]). For the uit disk graph the problem is kow to remai NP-hard; however, costat factor approximatios are possible i this case. For a recet survey o ad-hoc routig ad related problems, we refer to [7]. 3. NOTATION AND PRELIMINARIES I this sectio we itroduce otatios as well as some mathematical theorems which are used i the paper. The subject of this paper is the distributed costructio of domiatig sets of a etwork graph G = (V, E). For coveiece, we assume that V = {vl, v2,..., v~}, i.e. we assume that the etwork odes are labeled from to. These labels are ot used i our algorithms, but they simplify some proofs. By Ni, we deote the closed eighborhood of v~, i.e. Ni icludes v~ as well as all direct eighbors of vi. Where appropriate, Ni also deotes the set of the idices of the odes i N~. The degree of a ode v~ is called 5~ whereas A deotes the maximum degree i the etwork graph G. We will ofte make use of the maximum degree i a certai rage aroud a ode vi. For this purpose we defie 5~ ) ad 5~ 2) :,,~) :-~- maxsj, 5~2) := maxs!). jen i jen~ 3 Thus 5~) is the maximum degree of all odes i the closed eighborhood Ni of v~ whereas 5~2) is the maximum degree amog all odes at distace at most 2 from vi. For our algorithms, we use a purely sychroous model for commuicatio. That is, i every commuicatio roud, each ode is allowed to sed a message to each of its direct eighbors i G. I priciple, those messages ca be of arbitrary size; however, our algorithms oly use messages of size O(log A). We coclude this sectio by givig two facts which will the be used i subsequet sectios. Proofs are omitted ad ca be foud i stadard mathematical text books. FACT 3.. (Meas Iequality) Le.,4 C R + be a set of positive real umbers. The product of the values i.,4 ca be upper bouded by replacig each factor with the arithmetic mea of the elemets of A: ri -< x ~4 FACT 3.2. For > x >, we have 4. APPROXIMATING MDS BY LP RELAX- ATION The Miimum Domiatig Set (MDS) problem has bee itroduced i Sectio. I this sectio, we show how to obtai a I A approximatio by usig LP relaxatio techiques. For a itroductio to liear programmig see e.g. [4]. We first derive the iteger program which describes the MDS problem. Let S C V deote a subset of the odes of G. To each vi E V, we assig a bit xl such that x~ = ~ vi E S. For S to be a domiatig set, we have to demad that for each ode vi E V, at least oe of the odes i N~ is i S. Therefore, S is a domiatig set of G if ad oly if Vi E [,] : ~,eg.xj >. We defie the eighborhood matrix N to be the sum of the adjacecy matrix of G ad the <_ 26
idetity matrix (N is the adjacecy matrix with oes i the diagoal). The MDS problem ca the be formulated as a iteger program: subject to mi ~ xl i=l N - _x >_._ ~e {0,F. (IPMDs) By relaxig the coditio x E {0, } ~ to x > 0, we get the followig liear program: mi ~ xi i= subject to N x > x>o. (LPMDs) I the literature, the LP form of the domiatig set problem has also bee amed fractioal domiatig set problem. The correspodig dual liear program looks very similar to LPMDs: max ~ Yi i---- subject to N - y _<! _y _>_0. (DLPMDS) We have to assig a positive value yi to each ode vl. The sum of the y-values of the odes i the eighborhood Ni of a ode vi has to be less tha or equal to (for the correspodig x-values, this sum has to be greater tha or equal to ) ad the sum of all y-values, i.e. the objective fuctio has to be maximized. As a cosequece we get the followig lower boud o the size of a miimum domiatig set. LEMMA 4.. Let ~) be the maximum of the degrees of all odes i Ni as defied i Sectio 3. For ay domiatig set DS (i. e. also for a optimal oe), we have ~}a----y----- <-IDSI. i=l "- PROOF. Assigig yi := /(5~)+ ) yields a feasible solutio to the dual liear program DLPMDs. By the weak duality theorem, the value of the objective fuctio for ay feasible solutio for DLPMDs is smaller or equal to the value of the objective fuctio for ay feasible solutio for LPMDs. Hece, the objective fuctio for the DLPMDs-solutio is also smaller or equal to the size of ay domiatig set because ay feasible solutio for the iteger program IPMDs is feasible for LPMDs too. [] Let x* be a optimal solutio for LPMDs. Further let x (=) be a a-approximatio for LPMD$, i.e. x (=) is a feasible solutio for which _(4) < ~. ~ xt. (i) i=i i= I order to get a approximate solutio XDS for IPMDs from a a-approximatio x (s) for LPMDs, each ode applies the distributed Algorithm. Algorithm LPMDs ~ IPMDs Iput: feasible solutio _x (~) for LPMDs Output: IPMDs-Solutio X_D s (dom. set) : calculate 5~2) 2: pi := mi{, xi'(~), l(5~2)+ )} 0 with probability pl 3: XDS,i := otherwise 4: sed XDS# to all eighbors 5: if XDSj = 0 for all j E Ni the 6: XDS,i :---- 7: fi Remark: I lie 2, ~2) is calculated as follows. I a first roud, each ode vi seds its degree 5j to all eighbors. Afterwards 3~) (:= maxj~g~ ~k) is set to all eighbors i a secod roud. ~i~ 2) ca the be computed as maxj~ni ~). THEOREM 4.2. Let DSOPT be a miimum domiatig set ad let ~ be the greatest degree of the etwork graph G. ~'~) is a a-approximatio for LPMDs ad _XDs is the IPMDssolutio calculated by Algorithm with _x (~) as its iput. For the expected value of the size of the correspodig domiatig set DS (vi E DS ==~ xds,i = ), we have E [IDS[] < ( + al(a + ))-[DSoPT[. PROOF. A ode ca become a member of the domiatig set i lies 3 ad 6 of Algorithm. Let the radom variables X ad Y deote the umbers of odes which are selected i lies 3 ad 6, respectively. For the the expected value of X, we have IX] ---- Zpi <_ ~'~"~x!~) ) l(5~2)+, i=l i=0 <_ l(a + ) ~ ~i-(~) (zx_>6} 2)) i=i Eq. () i=o < a l(,~ + ). [DSoPT[. I order to compute the expected value of Y, we look at the probability ql that o ode i the direct eighborhood of ode vi (i.e. o ode i Ni) has bee selected. If ~j -(~) l(5~ 2)) ~_ for a vj E Ni, the correspodigpj = ad therefore qi = 0. Thus, we oly have to cosider the case 27
where all pj <. We obtai q~ = H(-pj) <_ H (- xj-(~)l(5})+ )) jeni jeni _< (l_~jen-(odl(5}l)-bl)) 5'+axiS~ + _< 5 + <_ e- ~(~?)+) 5} ) + The first iequality follows from 5} ) _< 5J 2), the secod iequality follows from Fact 3., the third iequality holds because _x (~) is feasible ad therefore the sum ~jen~ xj-('~) _>, ad the fourth iequality follows from Fact 3.2. For E [Y], we the have E[Y] = Eqi < () <- I DS PTI' ~= i= 5i + The last iequality follows from Lemma 4.. Addig E IX] ad E [Y] cocludes the proof. [] Remark : I lie 3 of Algorithm we could multiply xi with (l(5~(~) + ) - ll(5,(~ ) + )) istead of l(5~(~ ) + ). We would the obtai qi _< l(a + )/(5~ ) + ) ad the expected total size of the resultig domiatig set would be less tha or equal to 2a(l(A + ) - ll(a + )) IDSoPT [. Remark 2: Note that for regular graphs, Algorithm provides a very simple distributed algorithm to approximate MDS. Let the degree of each ode of a regular graph be 5. Assigig x~ := /(~ + ) for all odes vi yields a optimal solutio for LPMDs- Applyig Algorithm the results i a ( + l(5 + ))-approximatio for the MDS problem. I [6], Feige has prove that the domiatig set problem caot be approximated better tha by a approximatio ratio of la uless NP E DTIME( 0 gl g'~)) (up to lower order terms). Hece, uless NP almost equals P, the above algorithm is optimal whe applied to a optimal solutio of the LP relaxatio LPMDs of the domiatig set problem. However, the stregth of the approach of Algorithm lies i the potetial of distributig the calculatio over the odes of the etwork graph. Whe applied o a sigle computer, the greedy algorithm achieves the same approximatio ratio i time O(A) [8] while computig the liear program LPMDs with a iterior poit method would take sigificatly loger. I the ext sectio, we will show how to compute a approximatio of the liear program LPMDs usig a distributed algorithm. 5. APPROXIMATING THE LINEAR PRO- GRAM I this sectio, we preset the mai algorithm of this paper. We show how to fid a O(kA2/k)-approximatio of LPMDs i O(k 2) rouds. We will preset the algorithm i two variats. For the sake Of simplicity ad clarity, we will first preset a algorithm for the case that all odes kow the highest degree A i the etwork. I a secod step, we will the geeralize this algorithm such that the kowledge of A is ot ecessary ay more. Durig the algorithms, the odes icrease their x-values over time. I accordace with other domiatig set papers (e.g. [9, 0]), we say that a ode vl is colored gray as soo as the sum of the weights xj for vj E N~ exceeds, i.e. as soo as the ode is covered. Iitially all odes are colored white. The umber of white odes vj E N/at a give time is called the dyamic degree of vi ad deoted by 5(vi). Whe startig the algorithms, all odes are white, thus 5(vi) = 5/+. Assume ow that all odes kow A, the maximum degree of the etwork. Algorithm 2 is sychroously executed by all odes (a(vl) ad zi are auxiliary variables which are explaied later). Algorithm 2 LPMDs approximatio (A kow) : xi := 0; 2: for g := k- l to 0 by -l do 3: (* 5(v~) _< (A-.~....) (~+~)/k, z~ :=: 0 *) 4: form:=k- to0by- do 5: (, a(v~) < (zx + ) ('~+~)/~,) 6: sed color/ to all eighbors; 7: 5(vi) := [{j e Ni I colorj = 'white'}l; 8: if 5(vi) >_ (A + ) e/k the 9: ~ := max {~. ~x;~7=r~ ) 0: fi; : sed x/ to all eighbors; 2: if ~egxj > the colorl := 'gray' fi; 3: od 4: (. z./, < /(A + ) (~-j)/~ *) 5: od Before comig to a detailed aalysis of Algorithm 2, we give a geeral overview. Durig the algorithm, each ode v~ calculates the correspodig compoet xl of the solutio for LPMDs. Iitially all xi are set to 0, they are the gradually icreased as the algorithm progresses. The algorithm cosists of two ested loops. The purpose of the outer loop is to gradually reduce the highest dyamic degree i the etwork. As idicated by the ivariat i lie 3, 5(vl) is reduced by a factor (A + ) /k i every iteratio of the outer loop. I the ier loop, the x-values are icreased stepwise. By this we ca guaratee that the total weight is ot too high. Lemma 5. explais the ivariat of lie 3. LEMMA 5.. At the begiig of each iteratio of the outer loop off Algorithm 2, i.e. at lie 3, the dyamic degree 5(vl) of each ode vi is 5(vi) <_ (A + ) (e+l)/k. PROOF. For = k - the coditio reduces to 5(v/) < A + ad therefore follows from the defiitio of A. For all other iteratios the lemma is true because i the very last step of the precedig iteratio ( + ), all odes with 28
5(vi) >_ (A + ) (e+~)/~ have set x~ := i lie 9. By this all odes i Ni have tured gray ad therefore 5(vl) has become 0. Thus all degrees exceedig (zh + ) (~+l)/a have bee set to 0, for all others the ivariat already held beforehad. [] I a sigle iteratio of the outer loop, oly odes with ~(vi) > (A + ) e/~ icrease their x-value (lies 8-0). We call those odes active. The umber of active odes i the closed eighborhood Ni of a white ode vi at the begiig of a ier-loop iteratio (lie 5) is called a(vi). We defie a(vi) := 0 if vi is a gray ode. The purpose of the ier loop is to gradually reduce the maximum a(v) i the graph (ivariat i lie 5): LEMMA 5.2. At the begiig of each iteratio of the ier loop of Algorithm 2, i.e. at lie 5, a(vi) < (A+I) (m+l)/k for all odes v~ ~ V. PROOF. For m = k - we have a(vl) _< (zx + ) which is always true. For the other, cases, we prove that all odes vi with a(vi) too high have bee covered i the previous iteratio of the ier loop (i.e. they have become gray ad therefore a(vi) has become 0). We show that all odes vi for which a(vi) > (A + ) m/k at lie 5 are colored gray at the ed of the ier-loop iteratio (i.e. after lie 4). All active odes vj icrease xj such that xj _> /(zx+ ) m/k (lies 8-0 of Algorithm 2). If a(vi) > (A + ) m/~ there are more tha (A + ) m/k active odes i N~;. Therefore the sum of the x-values i N~ is greater or equal to after lie 0. [] I order to cout the weights assiged durig the iteratios of the ier loop, we assig a w~riable zi to each ode vi. I lie 3 all zi are set to 0. Wheever a ode vi icreases x~, the additioal weight is equally distributed amog the zj of all the odes vj i Ni which are white before the icrease of xi.. Hece the sum of the z-values is always equal to the sum of the x-icreases durig the curret iteratio of the outer loop. We ca show that at the ed of every iteratio of the outer loop, i.e. at lie 4, all zi < /(,~+ ) (~-l)/k. Together with the ivariat i lie 3, this eables us to prove a boud o the total weight of the additioal x-values i each iteratio of the outer loop. LEMMA 5.3. At the ed of a iteratio of the outer loop of Algorithm 2, i.e. at lie 4, for all odes vl E V. (A +-) PROOF. Because zi is set to 0 i lie 3, we oly have to cosider a sigle iteratio of the outer loop, i.e. a period i which g remais costat, zi ca oly be icreased as log as vi is a white ode. The icreases all happe i lie 9 because oly there the x-values are icreased. For each white ode vi, we divide the iteratio of the outer loop ito two phases. The first phase cosists of all ier-loop iteratios where vi remais white. The secod phase cosist of the remaiig ier-loop iteratios where vi becomes or is gray. Durig ~ the whole first phase ~jen~ xj <. Because all icreases of x-values are distributed amog at least (A + ) ~/k z-values we therefore get Zi < ~jen i Xj < (2) for phase. I lie 9 of the first ier-loop iteratio of the secod phase, z~ gets its fial value because oly z- values of white odes are icreased. All active odes have already bee active i the precedig ier-loop iteratio because 5(vj) ca oly become smaller over time. Thus from the precedig iteratio, all active odes vj E Ni have xj > /(A + ) (m+l)/k. I lie 9 they are ow icreased to /(A + ) '~/k. The differece of this value is distributed amog at least (~ + ) ~/k z-values ad because the umber of active odes i Ni is a(vl), the icrease of zi is at most (~+)'~ (a+l)z'~ a(vi). (3) (zh + )~: To obtai a boud o z~, we have to add its value before the icrease which is give by Equatio (2). From Lemma 5.2 we kow that a(v,) < (A + ) (m+l)/k. Pluggig this ito the sum of (2) ad (3), we obtai z~ _< (A + ) ¼ - +..._... (A + )~- (A+ )~ (A + )~ z' which cocludes the proof. [] We are ow ready to cosider the overall approximatioff ratio of Algorithm 2. THEOREM 5.4. For all etwork graphs G, Algorithm 2 computes a feasible solutio x for the liear program LPMDs such that x is a k(a + )2~k-approximatio of LPMDs. Further Algorithm 2 termiates after 2k 2 rouds. PROOF. For the umber of rouds, we see that each iteratio of the ier loop ivolves the sedig of two messages ad therefore takes two rouds. The umber of such iteratios is k 2. Further, the calculated x-values form a feasible solutio of LPMDs because i the very last iteratio of the ier loop (g = 0, m = 0) all odes vl with 5(vi) > set xi :=. This icludes all remaiig white odes. We prove the approximatio ratio of k(a + ) 2/k by showig that the additioal weight (i.e. sum of x-values) is upper-bouded by (A + ) 2/k i each iteratio of the outer loop. From Lemma 5., we kow that at lie 3, i.e. whe the iteratio starts, the dyamic degree 5(vl) of each ode vl is 5(vi) < (A + ) (t+l)/k. Hece there are at most (A + ) (t+l)/k o-zero z-values i the closed eighborhood of every ode v~ at the ed of a outer-loop iteratio at lie 4. Further Lemma 5.3 implies that all z-values are less tha or equal to (A + ) -(~-l)/k at lie 4. The sum of the z-values i the direct eighborhood of a ode vi durig each iteratio of the outer loop is 29
therefore upper-bouded by E z~ < (A+ )-~-- = (A+I)~. ~en, -- (ZX + ) -~-~ If we assig Yi := z~/(a + ) 2/k, the y-values form a feasible solutio for the dual LP DLPMDs because Vi : ~jen. Y~ <--. Hece the sum of all y-values is a lower boud o'the size of DSoPT ad therefore ~ z~ <_ (A + )u/aldsopw I for every iteratio of the outer loop. Because z is defied such that the sum over all z-values is equal to the sum over all icreases of the x-values, ad because there are k iteratios of the outer loop, we have i= _ < + ) ~ I DSoPTI. at the ed of Algorithm 2. [] The oly thig which caot be calculated locally i Algorithm 2 is the maximum degree ZX. Algorithm 3 is a adaptatio of Algorithm 2 where odes do ot eed to kow A. I each iteratio, Algorithm 3 assigs a xi which is greater or equal to the xi assiged i the correspodig iteratio of Algorithm 2. However, the xi are chose such that the approximatio ratio of k(a + ) 2/~ is preserved. Algorithm 3 LPMDs approximatio (A ot kow) : xi := 0; 2: calculate-i ~(2)., (. 2 corr.uicatio rouds *) 3: @2)(vi) := 5} 2) + ; 5(vi):= 5i + ; 4:for :=k- to0by-ldo 5: (* ~(vd < (~ + ) (~+~)I~, z~ := 0.) 6: form:=k- to0by- do 7: 2 if ~(vl) > 3 `()(vi) t+~ the 8: sed 'active ode' to all eighbors 9: fi; 0: a(vi) := I{J E Nilvj is 'active ode'}l; : if colori = 'gray' the a(vi) := 0 fi; 2: sed a(vi) to all eighbors; 3: a()(vi) := maxjen,{a(vj)}; 4: (, o.(,,d~ (~.(:~)(~) _< (~ + ) (m+~)/~ *) 5: 2 -Z-- if 5(vi) > 3`()(vi) ~+~ the 6: Xi := rax {xi,a()(vl) -~-f } 7: fi; 8: sed xi to all eighbors; 9: if ~je~v x~ >_ the colors := 'gray' fi; 20: sed colori to all eighbors; 2: 5(vi) := I{J e Ni I colors = 'white'} I 22: od; 23: (. zl _< (. ~- (zx -t-.):~/k)/@a)(v~) ~/(~+~) *) 24: sed 5(vi) to all eighbors; 25: @~)(vi) := maxjeg,{5(vj)}; 26: sed @l)(vl) to all eighbors; 27: @2)(vl) := max3en,{@~)(vj)} 28: od As for Algorithm 2, we first itroduce some otatio. 3`(d)(vi) deotes the maximum dyamic degree of all odes with distace at most d from vi at the begiig of the outer-loop iteratio. We use the otatio @d)(vl) istead of 5(a)(vi) because 3`(d)(vi) remais costat durig a iteratio of the outer loop while 5(vi) potetially chages after every iteratio of the ier loop. I each ier-loop iteratio, all odes which assig a ew x-value i lie 6 of Algorithm 3 are called active. As before, a(vi) deotes the umber of active odes i the direct eighborhood Ni of a white ode vi; for gray odes a(vi) := 0. a()(vi) is the maximum a(vj) amog all j E Ni. 5(v~) ad zi are used as i the previous algorithm. We are ow showig that Lemma 5. ad Lemma 5.2 (cf. Lemma 5.5 ad 5.6) also hold for Algorithm 3. LEMMA 5.5. At the begiig of each iteratio of the outer loop of Algorithm 3, i.e. at lie 5, the dyamic degree 5(v~) of each ode vi is 5(v~) <_ (A + ) (~+l)/k. PROOF. We use iductio to prove the lemma. Aalogously to Lemma 5., for the first iteratio (~ = k - ), the lemma follows from the defiitio of A. To prove the lemma for subsequet iteratios (iteratio step), we show that as for Algorithm 2, all odes with 5(v~) >_ (A+ ) t/k set x~ := i the last iteratio (m = 0) of the ier loop. Accordig to lies 5-7 of the algorithm, we see that all odes with 5(vi) > 3`(2)(vl)e/(e+l) set xi := for m = 0. Hece we have to show that Vi :@2)(vi)e/(e+l) < (2~ + ) e/k. By the iductio hypothesis, we kow that Vi :5(vl) _< (A + ) (e+l)/k at the begiig of the outer-loop iteratio. Because 3`(2)(vi) represets 5(vj) of some ode vj i the two-hop eighborhood of v~, we also have Vi : @2)(vi) < (A + ) (e+l)/k ad therefore 3`(2)(vi)d-~ < (zx + ) ~ '-~ = (~ + ) ~. LEMMA 5.6. Before assigig a ew value xi to vi i lies 5-7 of Algorithm 3, a(vi) _< (~ + ) (m+l)/k for all odes vi EV. PROOF. As for Lemma 5.2, we prove that all odes v/ for which a(vi) > (A + ) m/k at lie 4 are colored gray at the ed of the ier-loop iteratio (i.e. after lie 2). We use iductio over the iteratios of the ier loop. By the defiitio of A for every first iteratio of the ier loop (a(v~) < A + ) ad by the iductio hypothesis for all other iteratios, we have Vi : a(v~) < (A + ) (m+l)/k at lie 4. Therefore the weight each active ode vj assigs i lie 6 is xj >,. -- a()(vj),,,.+l -- (ZX + )"L:",:;' (A + ) ~' Because odes vl with a(vi) _> (A + ) m/k have at least (~+ ) talk active odes i the direct eighborhood, they are covered after each of their a(vi) eighbor odes vj assigs a weight xj > /(A + ) m/k. [] Lemma 5.7 is the aalogue to Lemma 5.3. [] 30
LEMMA 5.7. At lie 23 of Algorithm 3, for all odes vi ~ V. zi + (~x + ) PROOF. As i Algorithm 2, z~ is set to 0 at lie 5. Therefore, we oly have to cosider a sigle iteratio of the outer loop. Agai we cosider two phases. I the iteratios of the first phase vi remais white, the secod phase cosists of the iteratios where vi becomes or is gray. While the algorithm is i the first phase ~jen i Xj <. Further, all icreases of values xj are distributed amog at least 7(2)(Vj) e/oe+l) ~ "~()(Vi) /( +) z-values. Therefore, i aalogy to (2), we have (4) xj z~ < ~ 7(~)(~,~ < ~ (5 ~en~ 70)(vi) r~r for phase. I lie 6 of the first ier-loop iteratio of the secod phase, zi is chaged for the last time because oly z-values of white odes are icreased. There each active eighbor xj cotributes at most._l_ to the values zi. Because a(vl) _< ao)(v~) ad because vi has a(v~) active odes i the closed eighborhood N~ the total icrease of z~ is at most,,, e.a(vi) = a(vi) m+l ~. (6) a(vi)~z~ 7 ()(vi) ~. ~- 7()(Vl) TM By Lemma 5.6, we have a(v~) < (A + ) (m+~)/k durig a iteratio of the ier loop. Pluggig this ito (6) ad addig the value of z~ from the precedig iteratios (5) cocludes the proof: ((A + ):"~-"~-) "y+' + (A+ )¼ + Zi ~ [] As for the other algorithm, we aalyze each outer-loop iteratio separately to determie the approximatio ratio of Algorithm 3. By the defiitio of z, the sum of the x-values of a outer-loop iteratio is equal to the sum of the correspodig z-values. By Lemma 5.7 the sum of the z-values i the closed eighborhood of a ode v~ i a sigle iteratio of the outer loop is E zj < - I+(A+I)~ ~ _L_.~(~). (7) 3eg~ 7()(v~) ~+ Because 70)(vi) is the maximum dyamic degree i N~, 5(vl) < 70)(vi). Equatio (7) ca thus be formulated as Ezj~(l+(A+l)~)7('>(vi)~ --~. (8) jeni By Lemma 5.5 we kow that 7()(v~) < (A + ) (e+l)/k ad therefore 7()(vi)T-~ -f < (z~x"-b ) /k. Pluggig this ito Equatio (8) yields E zj <_ (LX+i) /k+(~+) 2/k. jeni By dividig all zi by the right had side of the above iequality, we obtai a feasible solutio for DLPMDs: Zi Yi:= (A+i)~+(A+i) ~ ~ Eyi_<l. jen~ The sum of the z-values of a outer-loop iteratio is therefore at most by a factor (A + ) /k + (A + ) 2/k larger tha the size of a optimal domiatig set. At the ed of the algorithm the sum over all xi (objective fuctio of LPMDs) is equal to the sum over the sums of the z~ for each outer loop iteratio. Therefore EXi ~k ((~ + j.)l/k..~ (z~ --~ )Z/k) IDSoPT[. i= [] THEOREM 5.8. For all etwork graphs G, Algorithm 3 computes a feasible solutio x with approximatio ratio k ((~X -l- ) /k -F (A + ) z/k) for the liear program LPMDs Further Algorithm 3 termiates after 4k z + O(k) rouds. PROOF. The ruig time (i.e. umber of rouds) ca be determied as for Algorithm 2. I each iteratio of the ier loop, 4 messages have to be set. This yields 4k 2 rouds for the totally k 2 ier-loop iteratios. There is a costat umber of additioal rouds i each outer-loop iteratio as well as at the begiig of the algorithm. Together, we get the claimed 4k 2 + O(k) rouds. Aalogously to Algorithm 2 x is feasible because i the very last iteratio of the ier loop (~ = 0, m = 0), all white odes vi set xi :=. Combiig Algorithms 3 ad i we obtai a distributed domiatig set algorithm. THEOREM 5.9. Applyig Algorithm 3 to obtai a LPMDsapproximatio ad Algorithm to covert this approximatio ito a domiatig set yields a distributed algorithm for the miimum domiatig problem which achieves a approximatio ratio of O(k~2/k log ~x) i O(k 2) rouds. PROOF. Theorem 5.9 directly follows from Theorems 4.2 ad 5.8. [] Remark: By settig k = e(log~x), we obtai a algorithm which computes a O(log 2 ~) approximatio for MDS i O(log z ~x) rouds. 3
6. CONCLUSION I this paper, we preseted a distributed approximatio algorithm for the miimum domiatig set problem. By computig a O(kA 2/k log A)-approximatio i O(k 2) rouds it is the first algorithm which achieves a o-trivial approximatio ratio i a costat umber of rouds. Particularly i the cotext of mobile ad-hoc etworks but also i more geeral etwork settigs, we believe that it is ofte advatageous to deploy algorithms which are very fast eve whe the calculated solutio is ot as good as the solutio of a less local algorithm. 7. ACKNOWLEDGMENTS We would like to thak Maurice Cochad, Juraj Hromkovi~, David Peleg, Peter Widmayer, ad Aaro Zolliger for fruitful discussios about the subject. 8. REFERENCES [] K. Alzoubi, P.-J. Wa, ad O. Frieder. Message-Optimal Coected Domiatig Sets i Mobile Ad Hoc Networks. I Proc. of the 3rd A CM It. Symposium o Mobile Ad Hoc Networkig ad Computig (MobiHOC), pages 57-64, EPFL Lausae, Switzerlad, 2002. [2] Y. Bartal, J. W. Byers, ad D. Raz. Global Optimizatio Usig Local Iformatio with Applicatios to Flow Cotrol. I Proc. of the 38th IEEE Symposium o the Foudatios of Computer Sciece (FOCS), pages 303-32, 997. [3] B. Berger, J. Rompel, ad P. Shor. Efficiet NC Algorithms for Set Cover with Applicatios to Learig ad Geometry. Joural of Computer ad System Scieces, 49:454-477, 994. [4] V. ChvAtal. Liear Programmig. W. H. Freema ad Compay, 983. [5] D. Dubhashi, A. Mei, A. Pacoesi, J. Radhakrisha, ad A. Sriivasa. Fast Distributed Algorithms for (Weakly) Coected Domiatig Sets ad Liear-Size Skeletos. I Proc. of the ACM-SIAM Symposium o Discrete Algorithms (SODA), pages 77-724, 2003. [0] L. Jia, R. Rajarama, ad R. Suel. A Efficiet Distributed Algorithm for Costructig Small Domiatig Sets. I Proc. of the 20th A CM Symposium o Priciples of Distributed Computig (PODC), pages 33-42, 200. [] D. S. Johso. Approximatio Algorithms for Combiatorial Problems. Joural of Computer ad System Scieces, 9:256-278, 974. [2] R. M. Karp. Reducibility Amog Combiatorial Problems. I Proc. of a Symposium o the Complezity of Computer Computatios, pages 85-03, 972. [3] S. Kutte ad D. Peleg. Fast Distributed Costructio of Small k-domiatig Sets ad Applicatios. Joural of Algorithms, 28:40-66, 998. [4] L. Lovasz. O the Ratio of Optimal Itegral ad Fractioal Covers. Discrete Mathematics, 3:383-390, 975. [5] M. Luby ad N. Nisa. A Parallel Approximatio Algorithm for Positive Liear Programmig. I Proc. of the 25th ACM Symposium o Theory of Computig (STOC), pages 448-457, 993. [6] S. Rajagopala ad V. Vazirai. Primal-Dual RNC Approximatio Algorithms for Set Cover ad Coverig Iteger Programs. SIAM Joural o Computig, 28:525-540, 998. [7] R. Rajarama. Topology Cotrol ad Routig i Ad hoc Networks: A Survey. SIGACT News, 33:60-73, Jue 2002. [8] P. Slav[k. A Tight Aalysis of the Greedy Algorithm for Set Cover. I Proc. of the 28th A CM Symposium o Theory of Computig (STOC), pages 435-44, 996. [9] J. Wu ad H. Li. O Calculatig Coected Domiatig Set for Efficiet Routig i Ad Hoc Wireless Networks. I Proc. of the 3rd It. Workshop o Discrete Algorithms ad Methods for Mobile Computig ad Commuicatios (DialM), pages 7-4, 999. [6] U. Feige. A Threshold of I for Approximatig Set Cover. Joural of the ACM (JACM), 45(4):634-652, 998. [7] J. Gao, L. Guibas, J. Hershberger, L. Zhag, ad A. Zhu. Discrete Mobile Ceters. I Proc. of the 7th aual symposium o Computatioal geometry (SCG), pages 88-96. ACM Press, 200. [8] M. R. Garey ad D. S. Johso. Computers ad Itractability, A Guide to the Theory of NP-Completeess. W. H. Freema ad Compay, 979. [9] S. Guha ad S. Khuller. Approximatio Algorithms for Coected Domiatig Sets. I Proc. of the ~th Aual Europea Symposium o Algorithms (ESA), volume 36 of Lecture Notes i Computer Sciece, pages 79-93, 996. 32